In this paper, we propose a second-order extension of the continuous-time game-theoretic mirror descent (MD) dynamics, referred to as MD2, which provably converges to mere (but not necessarily strict) variationally stable states (VSS) without using common auxiliary techniques such as time-averaging or discounting. We show that MD2 enjoys no-regret as well as an exponential rate of convergence towards strong VSS upon a slight modification. MD2 can also be used to derive many novel continuous-time primal-space dynamics. We then use stochastic approximation techniques to provide a convergence guarantee of discrete-time MD2 with noisy observations towards interior mere VSS. Selected simulations are provided to illustrate our results.

翻译：本文提出了一种连续时间博弈论镜像下降动力学（MD）的二阶扩展，称为MD2。该动力学在不依赖时间平均或折扣等常见辅助技术的情况下，可证明收敛到单纯（未必严格）的变分稳定状态（VSS）。我们表明，经过轻微修改后，MD2不仅具有无遗憾性质，还能以指数速率收敛到强VSS。MD2还可用于推导多种新颖的连续时间原始空间动力学。随后，我们利用随机逼近技术，为带噪声观测的离散时间MD2提供了收敛到内部单纯VSS的保证。最后通过数值模拟验证了相关结论。